## Small semi-weakly universal Turing machines (2007)

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Venue: | Machines, Computations and Universality (MCU), volume 4664 of LNCS |

Citations: | 9 - 4 self |

### BibTeX

@INPROCEEDINGS{Neary07smallsemi-weakly,

author = {Turlough Neary},

title = {Small semi-weakly universal Turing machines},

booktitle = {Machines, Computations and Universality (MCU), volume 4664 of LNCS},

year = {2007},

pages = {303--315},

publisher = {MCU, Springer. This Volume}

}

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### Abstract

Abstract. We present three small universal Turing machines that have 3 states and 7 symbols, 4 states and 5 symbols, and 2 states and 13 symbols, respectively. These machines are semi-weakly universal which means that on one side of the input they have an infinitely repeated word, and on the other side there is the usual infinitely repeated blank symbol. This work can be regarded as a continuation of early work by Watanabe on semi-weak machines. One of our machines has only 17 transition rules, making it the smallest known semi-weakly universal Turing machine. Interestingly, two of our machines are symmetric with Watanabe’s 7-state and 3-symbol, and 5-state and 4-symbol machines, even though we use a different simulation technique. 1.

### Citations

548 | A New Kind of Science
- Wolfram
(Show Context)
Citation Context ...g machines to date. Our semiweak machines are shown as solid diamonds and Watanabe’s as hollow diamonds. The standard and semi-weak machines are symmetric about the line where state = symbol. Wolfram =-=[26]-=- have found very small weakly universal machines which are illustrated as ∞ symbols in Figure 1. These weak machines simulate the cellular automaton Rule 110. Cook [3] proved (the proof is also sketch... |

79 |
Universality in Elementary Cellular Automata
- Cook
- 2004
(Show Context)
Citation Context ...machines that are plotted as hollow diamonds in Figure 1. A further generalisation are weak machines where we allow an infinitely repeated word to the left of the input and another to the right. Cook =-=[3]-=- andssymbol 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 0 ∞ � ∞ � � �� � �� ∞ ∞ � 6 : universal, bi-tag simulation, O(t ) �� 2 : semi-weakly universal, direct simulation, O(t ) � 4 2 : semi-weakly un... |

29 |
A Universal Turing Machine with Two Internal States. Automata Studies
- Shannon
- 1956
(Show Context)
Citation Context ...ine. Interestingly, our two machines are symmetric with Watanabe’s 7-state and 3-symbol, and 5-state and 4-symbol machines, even though we use a different simulation technique. 1 Introduction Shannon =-=[22]-=- was the first to consider the question of finding the smallest possible universal Turing machine, where size is the number of states and symbols. From the early sixties, Minsky and Watanabe had a run... |

28 | Small universal Turing machines
- Rogozhin
- 1996
(Show Context)
Citation Context ...sky’s technique of 2-tag simulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [19], Rogozhin =-=[21]-=-, Kudlek and Rogozhin [6], Baiocchi [1]. Neary and Woods [12,15] have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known Turing... |

20 |
Universality of Tag Systems With P = 2
- Cocke, Minsky
- 1964
(Show Context)
Citation Context ...10,11,23–25]. In 1962, Minsky [11] found a small 7-state, 4-symbol universal Turing machine. Minsky’s machine worked by simulating 2-tag systems, which where shown to be universal by Cocke and Minsky =-=[2]-=-. Rogozhin [20] extended Minsky’s technique of 2-tag simulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved... |

20 | P-completeness of cellular automaton rule 110
- Neary, Woods
(Show Context)
Citation Context ...r our machines is polynomial. More precisely, if M is a single tape deterministic Turing machine that runs in time t, then M is simulated by each of our semi-weak machines in time O(t 4 log 2 t). See =-=[13,14,27,28]-=- for further results and discussion related to the time complexity of small universal Turing machines. 1.1 Preliminaries All of the Turing machines considered in this paper are deterministic and have ... |

18 |
Three small universal Turing machines
- Baiocchi
- 2001
(Show Context)
Citation Context ...found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [19], Rogozhin [21], Kudlek and Rogozhin [6], Baiocchi =-=[1]-=-. Neary and Woods [12,15] have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known Turing machines, that obey the standard defin... |

18 |
Size and structure of a Universal Turing Machine using tag systems
- Minsky
- 1962
(Show Context)
Citation Context ...here size is the number of states and symbols. From the early sixties, Minsky and Watanabe had a running competition to see who could come up with the smallest machines [10,11,23–25]. In 1962, Minsky =-=[11]-=- found a small 7-state, 4-symbol universal Turing machine. Minsky’s machine worked by simulating 2-tag systems, which where shown to be universal by Cocke and Minsky [2]. Rogozhin [20] extended Minsky... |

16 | On the time complexity of 2-tag systems and small universal turing machines
- Neary, Woods
- 2006
(Show Context)
Citation Context ...r our machines is polynomial. More precisely, if M is a single tape deterministic Turing machine that runs in time t, then M is simulated by each of our semi-weak machines in time O(t 4 log 2 t). See =-=[13,14,27,28]-=- for further results and discussion related to the time complexity of small universal Turing machines. 1.1 Preliminaries All of the Turing machines considered in this paper are deterministic and have ... |

15 |
Frontier between decidability and undecidability: a survey
- Margenstern
- 2000
(Show Context)
Citation Context ...ese results to (semi-)weak machines with 1 state or 1 symbol. It is currently unknown if all lower bounds in Figure 1 generalise to (semi-)weak machines. It is also known from the work of Margenstern =-=[7]-=- and Michel [9] that the region between the non-universal curve and the smallest standard universal machines contains (standard) machines that simulate the 3x+1 problem and other related problems. The... |

15 | Four fast universal Turing machines
- Neary, Woods
- 2009
(Show Context)
Citation Context ...r our machines is polynomial. More precisely, if M is a single tape deterministic Turing machine that runs in time t, then M is simulated by each of our semi-weak machines in time O(t 4 log 2 t). See =-=[13,14,27,28]-=- for further results and discussion related to the time complexity of small universal Turing machines. 1.1 Preliminaries All of the Turing machines considered in this paper are deterministic and have ... |

13 | Four small universal Turing machines
- Neary, Woods
(Show Context)
Citation Context ...with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [19], Rogozhin [21], Kudlek and Rogozhin [6], Baiocchi [1]. Neary and Woods =-=[12,15]-=- have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known Turing machines, that obey the standard definition (deterministic, one... |

11 |
Small deterministic turing machines
- Kudlek
- 1996
(Show Context)
Citation Context ...standard Turing machine definition. The 1-symbol case is trivial, and the 1-state case was shown by Shannon [22] and, via another method, Hermann [4]. Pavlotskaya [16] and, via another method, Kudlek =-=[5]-=-, proved there are no universal 2-state, 2-symbol machines, where one transition rule is reserved for halting. Pavlotskaya [17] proved there are no universal 3-state, 2-symbol machines, and also claim... |

10 |
The uniform halting problem for generalized one state Turing machines
- Hermann
- 1968
(Show Context)
Citation Context ... the (even length) string of λ symbols, switching between states u1 and u2, to read the next dataword symbol in state u1: ⊢ 28 u1 u1 u1, . . .λ0λ1/000λ1/ 00000000000000000 000000000 0010010 0/0/0/... =-=(4)-=- The example is complete. ⊓⊔ The following example illustrates how U4,5 simulates the reading of 10 in the dataword. Specifically, the 10 is erased from the dataword, we append and erase the indexed a... |

10 |
A universal Turing machine with 3 states and 9 symbols
- Kudlek, Rogozhin
(Show Context)
Citation Context ...imulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [19], Rogozhin [21], Kudlek and Rogozhin =-=[6]-=-, Baiocchi [1]. Neary and Woods [12,15] have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known Turing machines, that obey the ... |

9 |
Minsky’s Small Universal Turing Machine
- Robinson
- 1991
(Show Context)
Citation Context ...0] extended Minsky’s technique of 2-tag simulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson =-=[19]-=-, Rogozhin [21], Kudlek and Rogozhin [6], Baiocchi [1]. Neary and Woods [12,15] have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smalle... |

8 |
Solvability of the halting problem for certain classes of Turing machines
- Pavlotskaya
- 1973
(Show Context)
Citation Context ...l curve in Figure 1 is shown for the standard Turing machine definition. The 1-symbol case is trivial, and the 1-state case was shown by Shannon [22] and, via another method, Hermann [4]. Pavlotskaya =-=[16]-=- and, via another method, Kudlek [5], proved there are no universal 2-state, 2-symbol machines, where one transition rule is reserved for halting. Pavlotskaya [17] proved there are no universal 3-stat... |

8 |
Dostatochnye uslovija razreshimosti problemy ostanovki dlja mashin T’juring. Problemi kibernetiki
- Pavlotskaya
- 1978
(Show Context)
Citation Context ...her method, Hermann [4]. Pavlotskaya [16] and, via another method, Kudlek [5], proved there are no universal 2-state, 2-symbol machines, where one transition rule is reserved for halting. Pavlotskaya =-=[17]-=- proved there are no universal 3-state, 2-symbol machines, and also claimed [16], without proof, that there are no universal 2-state, 3-symbol machines. Both cases assume that one transition rule is r... |

8 | 5-symbol 8-state and 5-symbol 6-state universal Turing machines - Watanabe - 1961 |

6 |
On the optimal number of instructions for universality of Turing machines connected with a finite automaton
- Margenstern, Pavlotskaya
- 2003
(Show Context)
Citation Context ...ly and semi-weakly universal machines, lend weight to the idea that finding non-universal lowerbounds in this region is difficult. For results on other generalisations of the Turing machine model see =-=[8,18]-=-, for example. Figure 1 shows our two new semi-weak machines as solid diamonds. These machines simulate cyclic tag systems, which were used [3] to show that Rule 110sis universal. It is interesting to... |

6 |
polynomial time universal Turing machines
- Small
(Show Context)
Citation Context ...with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [19], Rogozhin [21], Kudlek and Rogozhin [6], Baiocchi [1]. Neary and Woods =-=[12,15]-=- have recently found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known Turing machines, that obey the standard definition (deterministic, one... |

5 |
Turing machines and the generalized busy beaver competition
- Small
(Show Context)
Citation Context ...(semi-)weak machines with 1 state or 1 symbol. It is currently unknown if all lower bounds in Figure 1 generalise to (semi-)weak machines. It is also known from the work of Margenstern [7] and Michel =-=[9]-=- that the region between the non-universal curve and the smallest standard universal machines contains (standard) machines that simulate the 3x+1 problem and other related problems. These results, alo... |

5 |
Towards a precise characterization of the complexity of Universal and non-universal Turing Machines
- Priese
- 1979
(Show Context)
Citation Context ...ly and semi-weakly universal machines, lend weight to the idea that finding non-universal lowerbounds in this region is difficult. For results on other generalisations of the Turing machine model see =-=[8,18]-=-, for example. Figure 1 shows our two new semi-weak machines as solid diamonds. These machines simulate cyclic tag systems, which were used [3] to show that Rule 110sis universal. It is interesting to... |

5 |
Sem’ universal’nykh mashin Tjuringa
- Rogozhin
- 1982
(Show Context)
Citation Context ...In 1962, Minsky [11] found a small 7-state, 4-symbol universal Turing machine. Minsky’s machine worked by simulating 2-tag systems, which where shown to be universal by Cocke and Minsky [2]. Rogozhin =-=[20]-=- extended Minsky’s technique of 2-tag simulation and found small machines with a number of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [1... |

5 | 4-symbol 5-state universal Turing machine, Information Processing - Watanabe - 1972 |

4 |
A 6-symbol 7-state universal Turing machines
- Minsky
- 1960
(Show Context)
Citation Context ...ble universal Turing machine, where size is the number of states and symbols. From the early sixties, Minsky and Watanabe had a running competition to see who could come up with the smallest machines =-=[11, 12, 28, 29, 30]-=-. In 1962, Minsky [12] found a small 7-state, 4-symbol universal Turing machine. Minsky’s machine worked by simulating 2-tag systems, which where shown to be universal by Cocke and Minsky [3]. ∗ Damie... |

3 | On a minimal universal Turing machines - Watanabe - 1960 |

3 |
3N+1, UTM e tag-system
- Baiocchi
- 1998
(Show Context)
Citation Context ...or one-way infinite machines). It is currently unknown if all lower bounds in Figure 1 generalise to (semi-)weak machines. It is also known from the work of Margenstern [8], Michel [10], and Baiocchi =-=[1]-=- that the region between the non-universal curve and the smallest standard universal machines contains (standard) machines that simulate the 3x + 1 problem and other related problems. Kudlek [6] has g... |

3 |
Four small universal Turing machines, Fund
- Neary, Woods
(Show Context)
Citation Context ...ber of state-symbol pairs. Subsequently, some of Rogozhin’s machines were reduced in size or improved by Robinson [24], Rogozhin [26], Kudlek and Rogozhin [7], Baiocchi [2]. Recently, Neary and Woods =-=[13, 14, 15, 18]-=- have found small machines that simulate another variant of tag systems called bi-tag systems. All of the smallest known universal Turing machines, that obey the standard definition (deterministic, on... |